{"id":{"repo_id":"cambridge","oai_identifier":"oai:www.repository.cam.ac.uk:1810/325273"},"canonical_url":"https://search.dev.ndltd.org/etd/cambridge/oai:www.repository.cam.ac.uk:1810/325273","repository":{"repo_id":"cambridge","name":"Cambridge University","base_url":"https://api.repository.cam.ac.uk/server/oai/request"},"display":{"title":"Computing the Cassels-Tate Pairing for Jacobian Varieties of Genus Two Curves","abstract":"Let J be the Jacobian variety of a genus two curve defined over a number field K. The main focus of this thesis is on computing the Cassels-Tate pairing on the 2-Selmer group of J. We start by studying the Cassels-Tate pairing when J admits a Richelot 􏰑 isogeny φ : J → J. Suppose all points in J[2] are defined over K. We compute 􏰑􏰑 the Cassels-Tate pairing ⟨ , ⟩CT on Selφ􏰑(J)×Selφ􏰑(J) following the Weil pairing definition of the Cassels-Tate pairing. We then study the pairing ⟨ , ⟩CT on Sel2(J) × Sel2(J) following the homogeneous space definition of the Cassels-Tate pairing. For ε,η ∈ Sel2(J), we compute ⟨ε,η⟩CT both in the case where all points in J[2] are defined over K and in the case where the twisted Kummer surface Kη has a K-rational point. In both cases, we give a computable formula for ⟨ε,η⟩CT and a practical algorithm for computation when K = Q. In all cases, we calculate examples for which computing the Cassels-Tate pairing improves the rank bound of J obtained by carrying out standard de- scent calculations. We also give techniques to reduce the degree of the number field needed in the algorithm for computation.","abstract_html":"Let J be the Jacobian variety of a genus two curve defined over a number field K. The main focus of this thesis is on computing the Cassels-Tate pairing on the 2-Selmer group of J. We start by studying the Cassels-Tate pairing when J admits a Richelot 􏰑 isogeny φ : J → J. Suppose all points in J[2] are defined over K. We compute 􏰑􏰑 the Cassels-Tate pairing ⟨ , ⟩CT on Selφ􏰑(J)×Selφ􏰑(J) following the Weil pairing definition of the Cassels-Tate pairing. We then study the pairing ⟨ , ⟩CT on Sel2(J) × Sel2(J) following the homogeneous space definition of the Cassels-Tate pairing. For ε,η ∈ Sel2(J), we compute ⟨ε,η⟩CT both in the case where all points in J[2] are defined over K and in the case where the twisted Kummer surface Kη has a K-rational point. In both cases, we give a computable formula for ⟨ε,η⟩CT and a practical algorithm for computation when K = Q. In all cases, we calculate examples for which computing the Cassels-Tate pairing improves the rank bound of J obtained by carrying out standard de- scent calculations. We also give techniques to reduce the degree of the number field needed in the algorithm for computation.","abstract_has_math":false,"creators":["Yan, Jiali"],"institution":"University of Cambridge","degree_name":"Doctor of Philosophy (PhD)","degree_level":"Doctoral","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Fisher, Tom"],"committee_chairs":[],"committee_members":[],"year":2021,"date_issued":"2021-03-31","date_published":"2021-03-31","updated_at":"2026-07-22T22:24:24Z","subjects":["Number Theory","Cassels-Tate pairing","rank bound","genus two curves"],"languages":["eng"],"rights":[],"rights_urls":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/da1f8b48-8028-419b-9f6a-7562c548db32/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.17863/CAM.72729","outbound_label":"DOI","outbound_source":"dc:identifier.doi"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Fisher, Tom"]},{"key":"dc:creator","label":"Author","values":["Yan, Jiali"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.issued","label":"Date","values":["2021-03-31"]},{"key":"dc:publisher.institution","label":"Dc Publisher Institution","values":["University of Cambridge"]},{"key":"dc:relation.isreferencedby.uri","label":"Dc Relation Isreferencedby URI","values":["https://www.repository.cam.ac.uk/handle/1810/325273"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"dc:type.qualificationlevel","label":"Dc Type Qualificationlevel","values":["Doctoral"]},{"key":"dc:type.qualificationname","label":"Dc Type Qualificationname","values":["Doctor of Philosophy (PhD)"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Number Theory","Cassels-Tate pairing","rank bound","genus two curves"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language","label":"Dc Language","values":["eng"]},{"key":"dc:rights","label":"Dc Rights","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/da1f8b48-8028-419b-9f6a-7562c548db32/download","https://www.rioxx.net/licenses/all-rights-reserved/"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.doi","label":"DOI","values":["10.17863/CAM.72729"]},{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/f7785500-5f49-4b73-aba6-58cddc64f6c1/download"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Let J be the Jacobian variety of a genus two curve defined over a number field K. The main focus of this thesis is on computing the Cassels-Tate pairing on the 2-Selmer group of J. We start by studying the Cassels-Tate pairing when J admits a Richelot 􏰑 isogeny φ : J → J. Suppose all points in J[2] are defined over K. We compute 􏰑􏰑 the Cassels-Tate pairing ⟨ , ⟩CT on Selφ􏰑(J)×Selφ􏰑(J) following the Weil pairing definition of the Cassels-Tate pairing. We then study the pairing ⟨ , ⟩CT on Sel2(J) × Sel2(J) following the homogeneous space definition of the Cassels-Tate pairing. For ε,η ∈ Sel2(J), we compute ⟨ε,η⟩CT both in the case where all points in J[2] are defined over K and in the case where the twisted Kummer surface Kη has a K-rational point. In both cases, we give a computable formula for ⟨ε,η⟩CT and a practical algorithm for computation when K = Q. In all cases, we calculate examples for which computing the Cassels-Tate pairing improves the rank bound of J obtained by carrying out standard de- scent calculations. We also give techniques to reduce the degree of the number field needed in the algorithm for computation."]},{"key":"dc:format.checksum.md5","label":"Dc Format Checksum Md5","values":["7b72b0518cd287707a651e982db5210a","353adac0d1ebdfd65ab16480263c3c87"]},{"key":"dc:title","label":"Title","values":["Computing the Cassels-Tate Pairing for Jacobian Varieties of Genus Two Curves"]}]}],"canonical_facts":{"dc:contributor.advisor":["Fisher, Tom"],"dc:creator":["Yan, Jiali"],"dc:date.issued":["2021-03-31"],"dc:description.abstract":["Let J be the Jacobian variety of a genus two curve defined over a number field K. The main focus of this thesis is on computing the Cassels-Tate pairing on the 2-Selmer group of J. We start by studying the Cassels-Tate pairing when J admits a Richelot 􏰑 isogeny φ : J → J. Suppose all points in J[2] are defined over K. We compute 􏰑􏰑 the Cassels-Tate pairing ⟨ , ⟩CT on Selφ􏰑(J)×Selφ􏰑(J) following the Weil pairing definition of the Cassels-Tate pairing. We then study the pairing ⟨ , ⟩CT on Sel2(J) × Sel2(J) following the homogeneous space definition of the Cassels-Tate pairing. For ε,η ∈ Sel2(J), we compute ⟨ε,η⟩CT both in the case where all points in J[2] are defined over K and in the case where the twisted Kummer surface Kη has a K-rational point. In both cases, we give a computable formula for ⟨ε,η⟩CT and a practical algorithm for computation when K = Q. In all cases, we calculate examples for which computing the Cassels-Tate pairing improves the rank bound of J obtained by carrying out standard de- scent calculations. We also give techniques to reduce the degree of the number field needed in the algorithm for computation."],"dc:format.checksum.md5":["7b72b0518cd287707a651e982db5210a","353adac0d1ebdfd65ab16480263c3c87"],"dc:identifier.doi":["10.17863/CAM.72729"],"dc:identifier.uri":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/f7785500-5f49-4b73-aba6-58cddc64f6c1/download"],"dc:language":["eng"],"dc:publisher.institution":["University of Cambridge"],"dc:relation.isreferencedby.uri":["https://www.repository.cam.ac.uk/handle/1810/325273"],"dc:rights":["https://apollo8-f-pro.lib.cam.ac.uk/bitstreams/da1f8b48-8028-419b-9f6a-7562c548db32/download","https://www.rioxx.net/licenses/all-rights-reserved/"],"dc:subject":["Number Theory","Cassels-Tate pairing","rank bound","genus two curves"],"dc:title":["Computing the Cassels-Tate Pairing for Jacobian Varieties of Genus Two Curves"],"dc:type":["Thesis"],"dc:type.qualificationlevel":["Doctoral"],"dc:type.qualificationname":["Doctor of Philosophy (PhD)"]},"updated_at":"2026-07-22T22:24:24Z"}